First Cell Height Estimator

A simple tool to estimate the required first cell height based on a desired y+ value.

Disclaimer

This calculator is for educational and curiosity purposes only. No gaurantee on accuracy is inferred by a result.

Select Desired System of Units

Note: Use the US Standard Atmosphere Tool to determine properties of air at altitude.

You must enter a value for freestream velocity that is greater than 0.
You must enter a value for density that is greater than 0.
You must enter a value for dynamic viscosity that is greater than 0.
You must enter a value for boudary layer length that is greater than 0.
You must enter a value for Y+ that is at least 0.

Calculation Process

This approximation for a first cell height follows the process as outlined in several references regarding turbulence modeling. The process is well established and starts with the assumption that a desired value of y+ is known. This is typically based on the turbulence model being used in your numerical approach. The process is as follows.

  1. Calculate the Reynolds number for the flow field using the length of the boundary layer as the characteristic length term.
    \[Re = \frac{\rho V l}{\mu}\]
  2. Estimate the skin friction coefficient using one of the available models.
    Schlichting: \(C_f = {[2\log_{10}(Re_x)-0.65]^{-2.3}}\)   for \(Re_x < 10^9\)
  3. Calculate wall shear stress based on the result from above.
    \[\tau_w = C_f \: q_\infty \]
    where \(q_\infty\) is the dynamic pressure and is given by \[q_\infty = {1 \over 2} \rho V^2\]
  4. Calculate the friction velocity.
    \[u_* = \sqrt{\frac{\tau_w}{\rho}}\]
  5. Calculate the dimensional wall distance.
    \[y = \frac{y^+ \mu}{\rho \: u_*}\]